Inoue type manifolds and Inoue surfaces: a connected component of the moduli space of surfaces with K^2 = 7, p_g=0

We show that a family of minimal surfaces of general type with p_g = 0, K^2=7, constructed by Inoue in 1994, is indeed a connected component of the moduli space: indeed that any surface which is homotopically equivalent to an Inoue surface belongs to the Inoue family. The ideas used in order to show this result motivate us to give a new definition of varieties, which we propose to call Inoue-type manifolds: these are obtained as quotients \hat{X} / G, where \hat{X} is an ample divisor in a K(\Gamma, 1) projective manifold Z, and G is a finite group acting freely on \hat{X} . For these type of manifolds we prove a similar theorem to the above, even if weaker, that manifolds homotopically equivalent to Inoue-type manifolds are again Inoue-type manifolds.Comment: 36 pages, article dedicated to the 60-th birthday of Gerard van der Gee

The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes

We define a bounded cohomology class, called the {\em median class}, in the second bounded cohomology -- with appropriate coefficients --of the automorphism group of a finite dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the {\em median class of an action} by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to which extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show for example that any irreducible lattice in the product of at least two locally compact connected groups acts on a finite dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the Appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X. In the course of the proof we construct a \Gamma-equivariant measurable map from a Poisson boundary of \Gamma with values in the non-terminating ultrafilters on the Roller boundary of X.Comment: Minor changes that clarify some confusion have been made. Some figures have been adde